Confusion about Great Ellipse, Great Circle and Geodesic

[Master]

Hi.
I am a complete newbie in all of this and was recently trying to calculate distance between two gps coordinates.
I found geographiclib and started using it but the more I delve into this, the more confused I get.
Recently (as of couple of hours ago!) I tried to see how I can make sure a coordinate resides in a circle with a given radius. this very simple question got really confusing.
As I was searching for an answer I kept facing terms such as Great Circle Distance, Great Ellipse, Geodesic and more (like Great Ellipse sailing? I found it in this paper ).

So I knew I was using Geodesic all this time solely because the examples were using Geodesic! So I navigated to the documentation where it says :

The shortest path between two points on a ellipsoid at (lat1, lon1) and (lat2, lon2) is called the geodesic

so geodesic is the shortest path on an ellipsoid. ok so far so good. lets go on, then later on on the same page I read:

On a sphere, the geodesic is a great circle. On the ellipsoid, the problem is usually solved by transferring the geodesic to an "auxiliary sphere" where the latitude phi has been replaced by the reduced latitude beta where tan(beta) = (1 - f) tan(phi). On this sphere, the geodesic is again a great circle and the azimuth is the same as on the ellipsoid. However, the ellipsoidal distance is related to the great circle distance by an integral; and the ellipsoidal longitude is similarly related to the longitude on the auxiliary sphere

now I'm confused, I know based on this link great circle distance is used for intercontinental measurements :

The great circle distance is useful to evaluate the shortest path when intercontinental distances are concerned. It follows the sphericity of the globe, any shortest route is the one following the curve of the planet, along the parallels.
Because of the distortions caused by projections on a flat surface, a straight line on a map is not necessarily
the shortest distance. Ships and aircraft usually follow the great circle geometry to minimize distance and save time and money.

and then again here it talks about Great Ellipses and why they are inferior to geodesics now:

Great ellipses are sometimes proposed (Williams, 1996; Pallikaris & Latsas, 2009) as alternatives to geodesics for the purposes of navigation. This is predicated on the assumption that solving the geodesic problems is complex and costly. These assumptions are no longer true, and geodesics should normally be used in place of great ellipses

so geodesic which is defined on an ellipsoid is better than the great ellipses( which I still dont understand the difference from the ellipspoid that our geodesic is defined on), but where does great circle stand here?
how does great circle fare against great ellipses?
Haversine algorithm computes great circle distance(on a sphere) based on your answer here and it seems to be very fast and efficient and vincenty calculates the shortest path(ellipsoid of revolution which based on this is a spheroid)! this confuses me as I have seen a lot of answers on the internet that keep proposing Haversine for distance measurements, regardless of the distance .
I guess since its accurate for intercontinental distances and airplanes and ships use it, it can also be used for small distances?
If we use haversine, aren't we calculating a geodesic?
I have read your answer on stackoverflow, where you said choosing whether haversine or vincenty(which seems in geographiclib, geodesic is based on improved vincenty that you yourself have worked on?) is best for you depends on your application! apart from that, you also said:

For terrestrial applications, an ellipsoid of revolution is a reasonable approximation to "mean sea level"; the error is ± 100 m. The flattening of this ellipsoid is small, about 1/300, and so can be approximated by a sphere (of equal volume, for example).

Great circle distances differ from geodesic distances by up to 0.5%. In some applications, e.g., what's the distance from the Cape to Cairo?, this error can be neglected. In other applications, e.g., determining maritime boundaries, it is far too large (it's 5 m over a distance of 1 km). In general, you're safer using the geodesic distance.

What I take from this, is that I should be using geodesic instead of great circle/great ellipses.
However, what I don't understand is that why you said the error in geodesic is around 100m whereas you said here that
using geodesic we can achieve an accuracy of 12 nm using doubles.
I'm so confused!
if I want something accurate what should I be using? Why would I want to use one over the other? Is it solely because of the distance magnitude? or the altitude also affects this?
for example if I have a quadcopter that flies lets say around 100~200 meters high in the air, does the assumption concerning it the same as what is for an airplane that flies like in 500m~5000 meters? or should I just treat it like a car on the ground?

I'd greatly appreciate if you could shed some light on this.
Thank you in advance

[Charles Karney]

Use the geodesic distance. This can be used routinely to measure horizontal distances on the earth. Great ellipse and great circle distances are (sometimes poor) approximations to the geodesic distance. Note that these methods compute horizontal distances only. My comment about the "error being ±100 m" refers to how much the earth deviates (in the vertical direction) from a perfect ellipsoid.

[Master]

Thank you very much for your time and kind explanation. really appreciate it doctor.